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Your data matches 106 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [2]
=> 0
[2,1] => [1,1]
=> 1
[1,2,3] => [3]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [2,1]
=> 1
[3,1,2] => [2,1]
=> 1
[3,2,1] => [1,1,1]
=> 2
[1,2,3,4] => [4]
=> 0
[1,2,4,3] => [3,1]
=> 1
[1,3,2,4] => [3,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [3,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [3,1]
=> 1
[2,4,1,3] => [2,2]
=> 2
[2,4,3,1] => [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [2,1,1]
=> 2
[4,1,2,3] => [3,1]
=> 1
[4,1,3,2] => [2,1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> 3
[1,2,3,4,5] => [5]
=> 0
[1,2,3,5,4] => [4,1]
=> 1
[1,2,4,3,5] => [4,1]
=> 1
[1,2,4,5,3] => [4,1]
=> 1
[1,2,5,3,4] => [4,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> 2
[1,3,2,4,5] => [4,1]
=> 1
[1,3,2,5,4] => [3,2]
=> 2
[1,3,4,2,5] => [4,1]
=> 1
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [3,2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> 2
[1,4,2,3,5] => [4,1]
=> 1
[1,4,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [3,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> 2
[1,4,5,2,3] => [3,2]
=> 2
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000157
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(load all 6 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [3,1,2] => [[1,2],[3]]
=> 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> 1
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
=> 1
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 2
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000211
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> {{1}}
=> 0
[1,2] => [[1,2]]
=> {{1,2}}
=> 1
[2,1] => [[1],[2]]
=> {{1},{2}}
=> 0
[1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 2
[1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[2,4,3,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[3,2,4,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
[4,2,1,3] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[4,2,3,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
[4,3,1,2] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000228
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1] => [3]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> 1
[4,3,2,1] => [4]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000377
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 0
[1,2] => [2]
=> [1,1]
=> 1
[2,1] => [1,1]
=> [2]
=> 0
[1,2,3] => [3]
=> [1,1,1]
=> 2
[1,3,2] => [2,1]
=> [3]
=> 1
[2,1,3] => [2,1]
=> [3]
=> 1
[2,3,1] => [2,1]
=> [3]
=> 1
[3,1,2] => [2,1]
=> [3]
=> 1
[3,2,1] => [1,1,1]
=> [2,1]
=> 0
[1,2,3,4] => [4]
=> [1,1,1,1]
=> 3
[1,2,4,3] => [3,1]
=> [2,1,1]
=> 2
[1,3,2,4] => [3,1]
=> [2,1,1]
=> 2
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [2,2]
=> 1
[2,1,3,4] => [3,1]
=> [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> [4]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [3,1]
=> [2,1,1]
=> 2
[2,4,1,3] => [2,2]
=> [4]
=> 2
[2,4,3,1] => [2,1,1]
=> [2,2]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [2,2]
=> [4]
=> 2
[3,2,1,4] => [2,1,1]
=> [2,2]
=> 1
[3,2,4,1] => [2,1,1]
=> [2,2]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> 2
[3,4,2,1] => [2,1,1]
=> [2,2]
=> 1
[4,1,2,3] => [3,1]
=> [2,1,1]
=> 2
[4,1,3,2] => [2,1,1]
=> [2,2]
=> 1
[4,2,1,3] => [2,1,1]
=> [2,2]
=> 1
[4,2,3,1] => [2,1,1]
=> [2,2]
=> 1
[4,3,1,2] => [2,1,1]
=> [2,2]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [3,1]
=> 0
[1,2,3,4,5] => [5]
=> [1,1,1,1,1]
=> 4
[1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [4,1]
=> 2
[1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,2,5,4] => [3,2]
=> [5]
=> 3
[1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [3,2]
=> [5]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> 2
[1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,2,5,3] => [3,2]
=> [5]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [4,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> 2
[1,4,5,2,3] => [3,2]
=> [5]
=> 3
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000507
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [2]
=> [[1,2]]
=> 2 = 1 + 1
[2,1] => [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[1,2,3] => [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,4,3] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,2,4] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,3,4] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,3,4,1] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,2,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,1,2,3] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,1,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,1,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
[1,2,3,5,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,3,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,3,2,4,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,3,4,2,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,2,4] => [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,2,3,5] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,4,2,5,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000738
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,3,4,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,3,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,4,2,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,1,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,2,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,3,1,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 4 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,4,5,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,5,3,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,4,5,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,5,2,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,4,2,5,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,5,2,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000024
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000074
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 0
[1,2] => [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[2,1] => [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[1,2,3] => [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
[2,3,1] => [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
[3,1,2] => [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,2,4,3] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,3,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[2,1,3,4] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[2,3,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[2,4,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[3,2,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[3,4,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[4,1,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[4,1,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[4,2,1,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[4,3,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[1,2,3,5,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,2,4,3,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,2,4,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,2,5,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,3,2,4,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,3,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,3,4,2,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,3,5,2,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,4,2,3,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,4,2,5,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
[1,4,5,2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000141
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,3,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,1,3,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,3,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,4,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,2,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,1,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,1,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,2,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,3,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,4,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,4,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,5,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,3,2,4,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,3,4,2,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,5,2,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,2,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,4,2,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,5,2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
The following 96 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St000010The length of the partition. St000054The first entry of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000288The number of ones in a binary word. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St001726The number of visible inversions of a permutation. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000362The size of a minimal vertex cover of a graph. St000369The dinv deficit of a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000018The number of inversions of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001298The number of repeated entries in the Lehmer code of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000703The number of deficiencies of a permutation. St000470The number of runs in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000021The number of descents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000325The width of the tree associated to a permutation. St000051The size of the left subtree of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000956The maximal displacement of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. St000264The girth of a graph, which is not a tree. St001427The number of descents of a signed permutation. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001668The number of points of the poset minus the width of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St001896The number of right descents of a signed permutations. St001720The minimal length of a chain of small intervals in a lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001863The number of weak excedances of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000746The number of pairs with odd minimum in a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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